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| zuhair... |
 Posted: Thu Nov 05, 2009 4:53 pm |
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Guest
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Hi all,
Herein I want to set a rigorous definition of "Ad hoc" exposition of
a class theory.
The idea is that any class theory can be expositioned in an
ad hoc manner, I mean can be presented with an axiomatic system that
is ad hoc.
Now lets try to give a more precise meaning to what Ad hoc
presentation of a set theory would mean.
An axiom scheme is a single axiom of an axiom schema.
"An axiomatic system of a class theory T is said to be Ad hoc if :
"a proper subset of it's axiom schemes proves the existence of more
than one class;
OR
if there is two disjoint proper subsets of axiom schemes of T:
subset A and subset B, and Let "t" be a theorem proved from subset A
axioms without any axiom of subset B being involved in proving "t",
then if B and t can prove the existence of more than one class, then
the axiomatic system of T is ad hoc".
Examples: take Z for example, it is obvious that it is ad hoc
according to the first condition of the above definition.
However one may say that we can avoid the first condition of
ad hoc-ness by presenting Z in the following manner:
1.Extensionality
2.Separation
3.Union+pairing+power+infinity
However this doesn't avoid the second condition of ad hoc-ness, take
Union for example which is
For all y,z Exist x (y e x & z e x)
this can be derived as a theorem from axiom 3 alone without any of
axiom 1 or 2 being involved in proving it, now axiom 3 is disjoint
from axioms 1. and 2. Now the theorem of Union with axioms 1 and 2
would prove the existence of more than one class, thus this axiomatic
presentation of Z is also ad hoc.
I really don't know if one can show an alternative axiomatization of
Z and ZF such as to avoid being ad hoc according to the definition
above.
An example of a theory that is not Ad-hoc is NF.
Zuhair |
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| zuhair... |
| Posted: Thu Nov 05, 2009 4:55 pm |
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Guest
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Hi all,
Herein I want to set a rigorous definition of "Ad hoc" exposition of
a class theory.
The idea is that any class theory can be expositioned in an
ad hoc manner, I mean can be presented with an axiomatic system that
is ad hoc.
Now lets try to give a more precise meaning to what Ad hoc
presentation of a set theory would mean.
An axiom scheme is a single axiom of an axiom schema.
"An axiomatic system of a class theory T is said to be Ad hoc
if :
a proper subset of it's axiom schemes proves the existence of more
than one class;
OR
if there is two disjoint proper subsets of axiom schemes of T:
subset A and subset B, and Let "t" be a theorem proved from subset A
axioms without any axiom of subset B being involved in proving "t",
then if B and t can prove the existence of more than one class, then
the axiomatic system of T is ad hoc".
Examples: take Z for example, it is obvious that it is ad hoc
according to the first condition of the above definition.
However one may say that we can avoid the first condition of
ad hoc-ness by presenting Z in the following manner:
1.Extensionality
2.Separation
3.Union+pairing+power+infinity
However this doesn't avoid the second condition of ad hoc-ness,
take
Union for example which is
For all y,z Exist x (y e x & z e x)
this can be derived as a theorem from axiom 3 alone without any of
axiom 1 or 2 being involved in proving it, now axiom 3 is disjoint
from axioms 1. and 2. Now the theorem of Union with axioms 1 and 2
would prove the existence of more than one class, thus this axiomatic
presentation of Z is also ad hoc.
I really don't know if one can show an alternative axiomatization
of
Z and ZF such as to avoid being ad hoc according to the definition
above.
An example of a theory that is not Ad-hoc is NF.
Zuhair |
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| zuhair... |
| Posted: Thu Nov 05, 2009 5:59 pm |
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Guest
|
Hi all,
Herein I want to set a rigorous definition of "Ad hoc" exposition of
a class theory.
The idea is that any class theory can be expositioned in an
ad hoc manner, I mean can be presented with an axiomatic system that
is ad hoc.
Now lets try to give a more precise meaning to what Ad hoc
presentation of a set theory would mean.
An axiom scheme is a single axiom of an axiom schema.
"An axiomatic system of a class theory T is said to be Ad hoc
if:
a proper subset of it's axiom schemes proves the existence of more
than one class;
OR
if there is two disjoint proper subsets of axiom schemes of T:
subset A and subset B, and Let "t" be a theorem proved from subset A
axioms without any axiom of subset B being involved in proving "t",
then if B and t can prove the existence of more than one class, then
the axiomatic system of T is ad hoc".
Examples: take Z for example, it is obvious that it is ad hoc
according to the first condition of the above definition.
However one may say that we can avoid the first condition of
ad hoc-ness by presenting Z in the following manner:
1.Extensionality
2.Separation
3.Union+pairing+power+infinity
However this doesn't avoid the second condition of ad hoc-ness,
take Pairing for example which is
For all y,z Exist x (y e x & z e x)
this can be derived as a theorem from axiom 3 alone without any of
axiom 1 or 2 being involved in proving it, now axiom 3 is disjoint
from axioms 1. and 2. Now the theorem of Pairing with axioms 1 and 2
would prove the existence of more than one class, thus this axiomatic
presentation of Z is also ad hoc.
I really don't know if one can show an alternative axiomatization
of Z and ZF such as to avoid being ad hoc according to the definition
above.
An example of a theory that is not Ad-hoc is NF.
Zuhair |
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